What is the point of using 1D and 2D convolutions with a kernel size of 1 and 1x1 respectively?
Traditional CNNs used for image classification (and related tasks) are composed of 1 or more fully connected layers (FCs), after the convolutional and pooling layers, which take as input the features extracted from the convolutional and pooling layers, in order to perform classification or regression.
One problem with FCs in CNNs is that the number of parameters can be very big, with respect to the number of parameters in the convolutional layers.
There are tasks, such as image segmentation, where this big number of parameters is not really needed. An example of a neural network that does not make use of fully connected layers but only uses convolutions, downsampling (aka pooling), and upsampling operations is the U-net, which is used for image segmentation. A neural network that only uses convolutions is known as a fully convolutional network (FCN). Here I give a detailed description of FCNs and $1 \times 1$, which should also answer your question.
In any case, to answer your question more directly, $1 \times 1$ convolutions have been used for image segmentation tasks, i.e. dense classification tasks, i.e. tasks where you want to assign a label to each pixel (or a group of pixels), as opposed to sparse classification tasks such as image classification (where the goal is to assign 1 label to the whole image). Moreover, in comparison with FC layers, they have fewer parameters and, more importantly, the number of parameters in an FCN does not depend on the dimensions of the images (as in the case of traditional CNNs), which is a good thing (especially, when your images have high resolutions), but typically it depends on the number of kernels and instances (of objects), in the case of instance segmentation.
The FCN paper discusses this reduction of the number of parameters (and computation time), so you should probably read this paper for more details.